Resonance Occupation in the Kuiper Belt: Case Examples of the 5 : 2 and Trojan Resonances E

RESONANCE OCCUPATION IN THE KUIPER BELT: CASE EXAMPLES OF THE 5 : 2 AND TROJAN RESONANCES E. I. Chiang,1 A. B. Jordan,1 R. L. Millis,2 M. W. Buie,2 L. H. Wasserman,2 J. L. Elliot,2,3,4 S. D. Kern,3 D. E. Trilling,5 K. J. Meech,6 and R. M. Wagner7 Received 2003 January 21; accepted 2003 March 26

ABSTRACT As part of our ongoing Deep Ecliptic Survey (DES) of the Kuiper belt, we report on the occupation of the 1 : 1 (Trojan), 4 : 3, 3 : 2, 7 : 4, 2 : 1, and 5 : 2 Neptunian mean motion resonances (MMRs). The previously unrecognized occupation of the 1 : 1 and 5 : 2 MMRs is not easily understood within the standard model of resonance sweeping by a migratory Neptune over an initially dynamically cold belt. Among all resonant Kuiper belt objects (KBOs), the three observed members of the 5 : 2 MMR discovered by DES possess the largest semimajor axes (a 55:4 AU), the highest eccentricities (e 0:4), and substantial orbital inclinations (i 10 ). Objects (38084) 1999HB12 and possibly 2001KC77 can librate with modest amplitudes of 90 within the 5 : 2 MMR for at least 1 Gyr. Their trajectories cannot be explained by close encounters with Neptune alone, given the latter’s current orbit. The dynamically hot orbits of such 5 : 2 resonant KBOs, unlike hot orbits of previously known resonant KBOs, may imply that these objects were preheated to large inclination and large eccentricity prior to resonance capture by a migratory Neptune. Our ﬁrst discovered Neptunian Trojan, 2001QR322, may not owe its existence to Neptune’s migration at all. The trajectory of 2001QR322 is remarkably stable; the object can undergo tadpole-type libration about Neptune’s leading Lagrange (L4) point for at least 1 Gyr with a libration amplitude of 24. Trojan capture probably occurred while Neptune accreted the bulk of its mass. For an assumed albedo of 12%–4%, our Trojan is 130–230 km in diameter. Model-dependent estimates place the total number of Neptune Trojans resembling 2001QR322 at 20–60. Their existence helps to rule out violent orbital histories for Neptune. Key words: celestial mechanics — comets: general — Kuiper belt — minor planets, asteroids

1. INTRODUCTION and the ‘‘ Twotinos ’’ (2 : 1 resonant KBOs) are explored in detail by CJ. A fraction of Kuiper belt objects (KBOs) occupy low- We report here, as part of the ongoing survey of the order, exterior mean motion resonances (MMRs) estab- Kuiper Belt by the Deep Ecliptic Survey Team (Millis et al. lished by Neptune. Among the most well-known resonant 2002; Elliot et al. 2003), the previously unrecognized occu- KBOs are the Plutinos, which occupy the 3 : 2 MMR (Jewitt & Luu 2000). pation of the 5 : 2 and 1 : 1 (Trojan) Neptunian resonances. The three observed members of the 5 : 2 MMR stand out Plutinos have substantial orbital eccentricities, 0:1d among all resonant KBOs in having the largest semimajor ed0:3, an observation commonly interpreted to imply that a e Neptune migrated outward by several AU early in the his- axes ( 55:4 AU), the highest eccentricities ( 0:4), and substantial orbital inclinations (i 10). We will see that tory of the solar system (Malhotra 1995). The standard their dynamically hot orbits cannot be interpreted as ini- model of resonant capture and adiabatic excitation by a tially dynamically cold orbits that were modified purely by migratory Neptune predicts the 2 : 1, 5 : 3, 7 : 4, 3 : 2, and 4 : 3 resonance sweeping. Their existence points to another MMRs to be occupied by high-eccentricity objects dynamical excitation mechanism that likely operated prior (Malhotra, Duncan, & Levison 2000; Chiang & Jordan to Neptune’s migration. 2002, hereafter CJ). Occupation of the 4 : 3 MMR and possi- Our discovery of the first Neptunian Trojan librating bly of the 2 : 1 MMR has been reported by Nesvorny & Roig about the leading Lagrange (L4) point of Neptune vindi- (2001). Implications of Neptune’s migration for the Plutinos cates theoretical suggestions as to the long-term orbital stability of Neptunian Trojans (Holman & Wisdom 1993; Holman 1995; Gomes 1998; Nesvorny & Dones 2002). For 1 Center for Integrative Planetary Sciences, Department of Astronomy, University of California at Berkeley, 601 Campbell Hall 3411, Berkeley, example, Nesvorny & Dones (2002) find that about 50% of CA 94720; [email protected]. their hypothesized Neptunian Trojan population survives 2 Lowell Observatory, 1400 West Mars Hill Road, Flagstaff, AZ 86001. for 4 Gyr despite perturbations exerted by the other giant 3 Department of Earth, Atmospheric, and Planetary Sciences, planets. The stability of Neptune’s Trojan population con- Massachusetts Institute of Technology, 77 Massachusetts Avenue, trasts with the instability characterizing Saturnian and Cambridge, MA 02139. 8 4 Department of Physics, Massachusetts Institute of Technology, 77 Uranian Trojans on 10 yr timescales (Nesvorny & Dones Massachusetts Avenue, Cambridge, MA 02139. 2002; Gomes 1998). 5 Department of Physics and Astronomy, University of Pennsylvania, In x 2 we outline our procedure for identifying resonant David Rittenhouse Laboratory, 209 South 33rd Street, Philadelphia, KBOs in the face of observational uncertainties in their PA 19104. 6 Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822. orbits and describe the dynamical characteristics of our 5 : 2 7 Large Binocular Telescope Observatory, University of Arizona, and 1 : 1 resonant KBOs. Results of gigayear-long orbit Tucson, AZ 85721. integrations of our Trojan are presented at the end of this 430 RESONANCE OCCUPATION IN THE KUIPER BELT 431 section. In x 3 we briefly assess the plausibility of some theoretical scenarios that attempt to explain the observed pattern of resonance occupation. We consider models in which Neptune either sweeps objects into its resonances by virtue of its migration or populates the resonances by direct, violent gravitational scattering. A summary of our results, interpreted in the context of theoretical models, is provided in x 4.

2. OBSERVED RESONANCE MEMBERSHIP 2.1. Classification Procedure An object occupies an MMR if the resonant argument associated with that MMR librates. For the 5 : 2, e3 (third degree in the eccentricity of the KBO) Neptunian MMR, the argument is 5:2 ¼ 5 2N 3!~, where and !~ are the mean longitude and longitude of pericenter of the object, respectively, and N is the mean longitude of Neptune. For the 1 : 1, e0 Neptunian MMR, the argument equals 1:1 ¼ N. Testing for libration is a straightforward matter of integrating forward (or backward) the trajectory of an object in the gravitational fields of the Sun and the planets. A secure identification of a resonant KBO is made difficult by often substantial uncertainties in the initial position and velocity of the object, i.e., uncertainties in the osculating Keplerian ellipse fitted to astrometric observations. Bernstein & Khushalani (2000) derive a formalism for estimating these errors that is tailored for short-arc astrometric observations. Our Deep Ecliptic Survey (DES; Millis et al. 2002; Elliot et al. 2003) utilizes their formalism. Figures 1 and 2 depict the 1 and 2 confidence regions projected onto the a-e plane of our three 5 : 2 resonant can- didates and our one 1 : 1 resonant candidate. Uncertainties in a and e for these particular objects are small, of order 0.1%, thanks to the relatively extended, 1+ yr–long arcs of astrometry available for these KBOs. All elements reported in this paper are osculating, heliocentric elements referred to the J2000.0 ecliptic plane, evaluated at epoch JD 2,451,545.0. Surveying for libration in the six-dimensional confidence volume of possible initial orbits for each of hundreds of KBOs discovered by our DES is daunting. We proceed with a more limited agenda; in the six-dimensional error volume appropriate to a given KBO, we integrate, in addition to the nominal best-fit (initial) osculating orbit, two other solutions that lie on the 1 confidence surface and that have the Fig. 1.—Projections onto the a-e plane of the 1 and 2 confidence greatest and least semimajor axes. We refer to these sets of surfaces in the six-dimensional phase space of possible osculating orbits for initial conditions as orbit solutions 1, 2, and 3, respectively. 5 : 2 resonant KBOs (a) 1998WA31 and (b) 2001KC77. Solid black areas correspond to 1 confidence regions, while speckled areas correspond to The other five orbital elements are adjusted according to 2 confidence regions. The exact center of each plot corresponds to orbit their correlation with semimajor axis on the 1 confidence solution 1, while crosses denote orbit solutions 2 and 3. surface. We favor exploring the widest excursion in semimajor axis because that is the parameter that most influences resonance membership. The next most important parameter as of 2002 April 9 and given preliminary designations by the is eccentricity; however, as is evident in Figures 1 and 2, Minor Planet Center. We employ the regularized, mixed deviations in a and e are often strongly correlated, so that variable symplectic integrator, swift_rmvs3, developed by exploring the greatest deviation in a often implies that we Levison & Duncan (1994) and based on the N-body map of are also exploring the greatest deviation in e. Our choice of Wisdom & Holman (1991). We include the influence of the focusing on variations in a is further supported by the fact four giant planets, treat each KBO as a massless test par- that fractional errors in a (and e) are slower to converge to ticle, and integrate trajectories forward for 3 106 yr using zero than errors in i (Millis et al. 2002). a time step of 50 days, starting at JD 2,451,545.0. Initial We numerically integrate three sets of initial conditions positions and velocities for all objects are computed using for each of 204 KBOs discovered by the DES collaboration the formalism of Bernstein & Khushalani (2000) in the case 432 CHIANG ET AL. Vol. 126

Fig. 3.—Eccentricities, inclinations, and semimajor axes of resonant KBOs found in the DES. For all displayed objects, fractional 1 uncertainties in semimajor axis range from 0.003% to 3.5%, and orbit solutions 1, 2, and 3 yield consistent orbital classiﬁcations. Open diamonds represent resonant objects only; nonresonant objects will be presented by Elliot et al. (2003). Vertical lines indicate locations of nominal resonance with Neptune; dotted lines indicate uninhabited resonances, while dashed lines indicate inhabited resonances. Solid curves correspond to perihelion distances of 30, 35, 40, and 45 AU. Resonances with secure members include, in order of increasing distance from Neptune, the 1 : 1, 4 : 3, 3 : 2, 7 : 4, 2 : 1, and 5 : 2 MMRs.

displayed; by ‘‘ secure,’’ we mean that the 1 fractional uncertainties in semimajor axis are less than 10% and that all three sets of initial conditions give consistent orbit Fig. 2.—Projections onto the a-e plane of the 1 and 2 confidence classifications over 3 Myr. surfaces in the six-dimensional phase space of possible osculating orbits for A resonant object is one for which all three orbit solutions (38084) 1999HB12,a5: 2 resonant KBO, and 2001QR322,a1: 1 resonant yield libration of one or more of the same resonant argu- KBO. Solid black areas correspond to 1 confidence regions, while speckled areas correspond to 2 confidence regions. The exact center of ments; nonresonant objects exhibit no libration of any reso- each plot corresponds to orbit solution 1, while crosses denote orbit nant argument among the three solutions. solutions 2 and 3. The locations of the points in Figure 3 correspond to the semimajor axes, eccentricities, and inclinations at the start of the integration. Error contours are much smaller than the of short-arc orbits and from E. Bowell’s database in the case sizes of the symbols in most cases. Dotted lines delineate the of long-arc orbits (see Millis et al. 2002). The relative energy locations of nominal resonance with Neptune. In addition error over the integration is bounded to less than 107.A to confirmed librators in the 4 : 3, 3 : 2, 7 : 4, and 2 : 1 total of 107 different resonant arguments are examined for resonances, the 1 : 1 and 5 : 2 resonances contain one and libration. Full details of our procedure are provided in three members, respectively. Elliot et al. (2003). Figure 3 displays only objects discovered by the DES col- Some results of this procedure are showcased in Figure 3, laboration. Other groups have reported occupation of the which contains only a small subset of the data to be released 3 : 2, 4 : 3, and 2 : 1 resonances. For example, Nesvorny & by Elliot et al. (2003). Only ‘‘ secure ’’ resonant objects are Roig (2001) have reported KBOs occupying the 4 : 3 MMR No. 1, 2003 RESONANCE OCCUPATION IN THE KUIPER BELT 433

TABLE 1 Designations of DES and Non-DES Resonant KBOs

Resonance Name 1 : 1...... 2001QR322 5 : 4...... 1999CP133 4 : 3...... 1998UU43, 2000CQ104, (15836) 1995DA2 3 : 2...... (28978) Ixion*, 1998UR43, 1998US43, 1998WS31, 1998WU31, 1998WV31, 1998WW24, 1998WZ31, 2000CK105, 2001KY76, 2001KB77, 2001KD77, 2001KQ77, 2001QF298, 2001QG298, 2001RU143, 2001RX143, (15788) 1993SB, (15789) 1993SC, (15810) 1994JR1, (15820) 1994TB, (15875) 1996TP66, (19299) 1996SZ4, (20108) 1995QZ9, (24952) 1997QJ4, (32929) 1995QY9, (33340) 1998VG44, (38628) 2000EB173, (47171) 1999TC36, (47932) 2000GN171, 1993RO, 1995HM5, 1996RR20, 1996TQ66, 1998HH151, 1998HK151, 1998HQ151, 1999CE119, 1999CM158, 1999TR11, 2000FV53, 2000GE147, 2001FL194, 2001VN71, 2001YJ140, 2002VE95, 2001FU172 5 : 3...... (15809) 1994JS, 1999CX131, 2001XP254 7 : 4...... 2000OP67, 2001KP77, 1999KR18, 1999RH215, 2000FX53, 2000OY51 2 : 1...... 2000QL251, (20161) 1996TR66, (26308) 1998SM165, 1997SZ10, 1999RB216, 2000JG81 7 : 3...... 1999CV118 5 : 2...... (38084) 1999HB12, 1998WA31, 2001KC77, (26375) 1999DE9, 2000FE8, 2000SR331, 2002TC302 Note.—Objects discovered by DES are denoted by an asterisk. and possibly the 2 : 1 MMR. When we integrate the trajecto- What is the long-term evolution of these objects? We have ries of non-DES objects, we confirm the results of Nesvorny integrated orbit solutions 1, 2, and 3 for all three objects for- & Roig (2001) that 1995DA2 inhabits the 4 : 3 MMR and ward by 1 Gyr. For (38084) 1999HB12, all three orbit solu- that (20161) 1996TR66 and 1997SZ10 inhabit the 2 : 1 tions yield libration in the 5 : 2 resonance for the full MMR. The names of resonant objects discovered by the duration of the integration. The same is true for orbit solu- DES team and by non-DES teams are contained in Table 1. tion 2 of 2001KC77. For the aforementioned four trajecto- Whereas the 4 : 3, 3 : 2, 7 : 4, and 2 : 1 resonances are pre- ries, the libration amplitudes range from 90 to 100.By dicted by the standard migration model for Neptune to be contrast, solution 1 of 2001KC77 eventually yields circula- substantially populated (see Figs. 3 and 4 of CJ), the 5 : 2 tion, while solution 3 leads to a close encounter with and 1 : 1 resonances are not. We focus our attention now on Neptune 0.512 Gyr into the simulation. For 1998WA31,all the newly discovered members of the 5 : 2 and 1 : 1 MMRs, three solutions eventually end with a close encounter with to investigate the constraints they place on the dynamical Neptune, with solution 2 lasting the longest (0.882 Gyr). We history of the Kuiper belt. conclude that among our three 5 : 2 resonant members, (38084) 1999HB12 and possibly 2001KC77 are likely to be 2.2. Observed Members of the 5 : 2 MMR long-term and therefore primordial residents of the 5 : 2 MMR. Note further that the accuracy of our orbital solu- Evolutions of the resonant argument, 5:2, for objects tion is highest for (38084) 1999HB12 and lowest for 1998WA31, (38084) 1999HB12, and 2001KC77 are displayed in Figure 4. The integrations shown begin with nominal best-fit initial conditions; the other two sets of initial conditions yield nearly identical results. The resonant argument, 5:2, librates in the manner shown in Figure 4 for the entire duration of the integration, 3 Myr. The libration centers are h5:2i¼180 , the amplitudes are D5:2 max 5:2 h5:2i90 140 , and the libration periods are Tl 2 104 yr. The libration period increases with decreasing libration amplitude, unlike the case for the conventional pendulum model for a resonance. For each object, we further explore error space by integrating eight additional sets of initial conditions that lie on the 2, 3, 4, and 5 confidence surfaces and that are characterized by semimajor axes that deviate most from the best- fit semimajor axis in positive and negative senses. Objects 3 (38084) 1999HB12 and 2001KC77 remain in the 5 : 2 e resonance in all cases for 3 Myr. We conclude that our identifications of (38084) 1999HB12 and 2001KC77 as current 5 : 2 librators are particularly secure. Object 1998WA31 fails to librate in the 5 : 2 resonance when its initial semimajor axis is less than the nominal value by 2 or more, i.e., when a is less than the nominal value by more than 0.13 AU. How- ever, other sets of initial conditions for which a is greater than the nominal value yield libration even at the 5 level for 1998WA31. Our identification of 1998WA31 as a current Fig. 4.—Libration of the resonant argument, 5:2, for our observed member of the resonance is therefore less firm than for the members of the 5 : 2 resonance. Integrations begin with nominal best-fit others, but not alarmingly so. initial conditions (orbit solution 1). 434 CHIANG ET AL. Vol. 126

TABLE 2 Orbital Elements of 5 : 2and1: 1 Resonant KBOs

a i ! M Name Resonance (AU) e (deg) (deg) (deg) (deg)

1998WA31 ...... 5 : 2 55.73 0.432 9.43 20.7 310.7 28.2 (38084) 1999HB12...... 5 : 2 55.10 0.409 13.17 166.5 66.7 343.1 2001KC77...... 5 : 2 54.67 0.352 12.9 57.8 181.8 358.4 2001QR322 ...... 1 : 1 30.39 0.028 1.32 151.6 236.2 327.3

Note.—Osculating, heliocentric elements referred to the J2000.0 ecliptic plane, evaluated at epoch JD 2,451,545.0. Elements shown here are best-ﬁt values; for a discussion of uncertainties, see x 2.1 and Elliot et al. 2003. The angles , !, and M are the longitude of ascending node, the argument of perihelion, and the mean anomaly, respectively.

1998WA31; thus, it seems possible that with more astrome- tested, object 2001QR322 librates in the 1 : 1 MMR. We tric measurements, all three objects will be found to stably regard our identification of 2001QR322 as a Neptunian occupy the 5 : 2 MMR over gigayear-long timescales. Trojan as extremely secure. Orbital elements for our three 5 : 2 resonant KBOs are The trajectory of the Trojan in the Neptune-centric frame provided in Table 2. is showcased in Figure 6. A tadpole-like path whose center is shifted forward in longitude from Neptune’s L4 point is 2.3. Observed Member of the 1 : 1 MMR evident; the longitude shift of 5 is expected for finite amplitude librators (see, e.g., Murray & Dermott 1999, their The evolution of the resonant argument, 1:1, for object Fig. 3.11). The minimum distance of approach to Neptune 2001QR322 is displayed in Figure 5. Only the integration of over 3 Myr is approximately 20 AU. the best-fit solution is shown; orbit solutions 2 and 3 yield Orbital elements for our Trojan are listed in Table 2. Note nearly identical results. All three sets of initial conditions that the orbital elements of 2001QR322 lie consistently yield tadpole-type libration about Neptune’s L4 point for at within the region of 4 Gyr–long stability mapped by least 1 Gyr. The libration center is h1:1i64=5, the libra- Nesvorny & Dones (2002; see their Fig. 9c). Our object is a tion amplitude is D1:1 24 , and the libration period is member of the low-inclination population of stable 4 Tl 10 yr. Neptunian Trojans; Nesvorny & Dones (2002) find surpris- We further explore error space by integrating eight ingly that Neptunian Trojans having orbital inclinations as additional solutions that deviate from the nominal best-fit high as 25 are also stable. Note further that the libration solution by 2, 3, 4, and 5 , each for 3 Myr. In all cases amplitude of 2001QR322 (24 ) also lies consistently below the stability threshold of 60–70 established by Nesvorny & Dones (2002). How many Neptunian Trojans might there be in all? Nesvorny & Dones (2002) provide three models of the sky density of Neptune Trojans that differ in the assumed distribution of orbital elements. We have combined their models (see their Figs. 11, 13, and 14) with the distribution of our DES search fields to estimate that 20, 60, and 40 Neptune Trojans having diameters and albedos comparable to those of 2001QR322 exist in all, based on their models I, II, and III, respectively. The above numbers already include Trojans librating about Neptune’s L5 point, which we assume to have the same population as L4 librators. While it is impossible to differentiate between the models based only on the discovery of a single object, it is heartening to see that all three models give the same order-of-magnitude estimate for the total number of Neptune Trojans resembling 2001QR322.

3. THEORETICAL IMPLICATIONS OF RESONANCE OCCUPATION Here we briefly explore theoretical implications of the observed occupation of the 5 : 2 and 1 : 1 Neptunian MMRs. We aim, in particular, to test the hypothesis that Neptune Fig. 5.—Evolution of the resonant argument, 1:1 ¼ N, for our migrated outward by several AU during the solar system’s Neptunian Trojan, based on best-fit orbit solution 1. The object remains past and, in so doing, sculpted the pattern of resonance bound to the 1 : 1 resonance for at least 1 Gyr and betrays no sign of instability. Top and bottom panels display the same evolution with different occupation in the Kuiper belt (Fernandez & Ip 1984; time resolutions. Orbit solutions that deviate from the best-fit solution by Malhotra 1995; CJ). Sections 3.1 and 3.2 focus on the 5 : 2 as much as 5 also yield libration for at least 3 Myr (data not shown). MMR, while x 3.3 is devoted to the 1 : 1 MMR. No. 1, 2003 RESONANCE OCCUPATION IN THE KUIPER BELT 435

Fig. 6.—Trajectory of 2001QR322, our Neptunian Trojan, in a quasi–Neptune-centric frame. The left-hand panel displays a bird’s-eye view of the outer solar system, with the giant planet orbits shown schematically. The dark tube of points lying on Neptune’s orbit marks the computed path of the Trojan. The length of the vector from the origin to each point on the tube gives the instantaneous heliocentric distance of the object; the angle between this vector and the abscissa gives the instantaneous angle between the Sun-Trojan and Sun-Neptune vectors. The Trojan librates along Neptune’s orbit as indicated by the solid and dotted curved arrows. Each libration takes about 104 yr to complete. The small inset rectangle is magniﬁed in the right-hand panel to show the fast epicyclic motion. Each fast epicycle takes about one orbital period of Neptune, or about 200 yr, to complete.

3.1. Neptune’s Migration and the 5 : 2MMR adopt values for the initial and final semimajor axes, (ai, af), for each of the planets as follows (in AU): Jupiter (5.40, Could KBOs in the 5 : 2 resonance have been trapped into 5.20), Saturn (8.78, 9.58), Uranus (16.2, 19.2), and Neptune that MMR as it swept across the Kuiper belt? We consider (23.1, 30.1). We employ the symplectic integrator, SyMBA two scenarios, one in which Neptune migrates into a sea of (Duncan, Levison, & Lee 1998), as kindly supplied to us by initially dynamically cold test particles, and another in E. Thommes. We adopt a time step of 0.6 yr. For more which the planet migrates into a sea of initially dynamically details, the reader is referred to CJ. hot particles. Note that our simulations prescribe the migration to be smooth. If the planetesimals that scattered off Neptune and 3.1.1. Cold Initial Conditions drove its migration were sufficiently massive, our idealization For objects on initially low-eccentricity orbits, the proba- would be invalid. We estimate that our approximation is bility of capture into the 5 : 2, third-order resonance is pro- valid if most of the mass of the planetesimal disk were con- hibitively small compared to the probability of capture into tained in bodies having radii less than 40 km. The deriva- low-order resonances such as the 2 : 1 and 3 : 2. Neither the tion of our crude estimate is contained in the Appendix. The simulations performed by Malhotra et al. (2000) nor those actual sizes of ancient planetesimals scattering off Neptune by CJ report any object caught into the 5 : 2 resonance are, of course, unknown, although Kenyon (2002) calculates among the 100 test particles over which that resonance in his accretion simulations that 90% of the solid mass at swept. We have executed another migration simulation, fol- heliocentric distances of 40–50 AU in the primordial solar lowing those of CJ, that is tailored to gauge the capture effi- system may be contained in 0.1–10 km–sized objects. ciency of the 5 : 2 resonance for objects on initially nearly Figure 7 demonstrates that capture into the 5 : 2 reso- circular, low-inclination orbits. The simulation parameters nance, even when Neptune takes as long as a few times 107 are identical to those in CJ’s model I, except that the initial yr to migrate outward by several AU, is improbable; only semimajor axes of the 400 test particles range from 43.55 one out of 400 objects librates in the 5 : 2 resonance at the AU (=1 AU greater than the initial location of the 5 : 2 reso- end of the simulation. By contrast, the 2 : 1 resonance boasts nance) to 54.44 AU (=1 AU less than the final location of 90 captured objects. The predicted population ratio between the 5 : 2 resonance). Thus, all such objects are potentially the 5 : 2and2: 1 resonances is not easily reconciled with the swept into the migrating 5 : 2 resonance. Their initial eccen- observations as depicted in Figure 3. Accounting for the tricities and inclinations are randomly and uniformly dis- observational bias in favor of finding 2 : 1 members over tributed between 0.00 and 0.05 and between 0.00 and 0.025 5 : 2 members as a result of the fact that the 5 : 2 resonance is rad, respectively. Arguments of periastron (!), longitudes of more distant than the 2 : 1 would only accentuate the dis- ascending nodes (), and mean anomalies (M) are uni- agreement. When both the effects of greater distance and formly and randomly sampled between 0 and 2. The semi- differential longitudinal clustering of resonant KBOs are major axis of each giant planet evolves with time, t, accounted for, we estimate that bias correction factors of according to 3 in favor of finding 2 : 1 members result (see CJ for a dis- t cussion of how these bias corrections are estimated). Even if aðtÞ¼af af ai exp ; ð1Þ the difference between the predicted 1 : 90 ratio and the observed 3 : 1 ratio were to be attributed to extremely strong where we fix the migration timescale, ,tobe107 yr. We and positive radial gradients in the primordial surface 436 CHIANG ET AL. Vol. 126

Fig. 7.—Results of a migration simulation designed to gauge the capture efficiency of the 5 : 2 resonance. Left-hand panels display cold initial conditions of 400 test particles prior to sweeping by the 5 : 2 resonance, whose nominal location is indicated by the dashed line. Right-hand panels display the aftermath of resonance sweeping, where open diamonds denote resonantly librating particles. Only one particle is caught by the 5 : 2 MMR; its eccentricity is pumped to 0.16, and its inclination is relatively unaltered. By contrast, 90 particles are swept into the 2 : 1 resonance at a 47:8 AU. The ratio of 1 : 90 is difficult to reconcile with the observed 3 : 1 ratio showcased in Fig. 3; accounting for the bias introduced by the fact that the 5 : 2 MMR is more distant than the 2 : 1 would only worsen the disagreement. Moreover, the predicted final e and i of our simulated 5 : 2 resonant object are much lower than observed values. density of planetesimals (an unnatural prospect in itself), formly and randomly distributed between 0 and 0.3 and resonant excitation by the 5 : 2 MMR of initially cold orbits between 0 and 0.15 rad, respectively. The result is summar- would result in eccentricities and inclinations that are ized in Figure 9. Of 400 particles potentially caught by the generally much too low compared with the observations. sweeping 5 : 2 MMR, 20 are captured and have their eccen- Finally, we note that when non-DES and DES data sets tricities amplified to final values of 0.2–0.5. We have verified are combined, the number of securely identified 2 : 1and 5 : 2 that these 20 objects represent adiabatic capture events and resonant KBO increase to 6 and 7, respectively. Blithely not violent scatterings; their semimajor axes increase using these numbers, which are affected by observational smoothly over the duration of the simulation from values of biases from non-DES surveys that we have to not quanti- as low as 45 AU to the final resonant value of 55.4 AU. fied, still yields a ratio (7 : 6) that is hard to reconcile with In addition, five objects are adiabatically swept into the the predicted ratio (1 : 90) 3 : 1 resonance whose final location lies at a 62 AU. Are the predicted libration amplitudes consistent with those observed? The answer is yes; libration amplitudes of 3.1.2. Hot Initial Conditions our three observed KBOs range from 90 to 140, while Figure 8 displays the width of the 5 : 2 resonance in a-e those of our 20 simulated 5 : 2 resonant particles range from space. Since the resonance widens considerably at ee0:2, it 16 to 145, with six particles having amplitudes in the is worth considering whether the capture efficiency increases observed range. with increasing initial eccentricity. That the Kuiper belt has While the capture efficiencies of high-order MMRs such been disturbed by more than the (hypothesized) slow sweep- as the 5 : 2 and 3 : 1 resonances magnify with increasing ini- ing of Neptune’s MMRs is evidenced by KBOs’ large orbi- tial eccentricity, those of low-order MMRs such as the 3 : 2 tal inclinations (Brown 2001; CJ). and 2 : 1 resonances decrease. In the simulation just We repeat the migration simulation of x 3.1.1 but with described, 29 objects are caught in the 2 : 1 resonance—a initial eccentricities and inclinations of test particles uni- factor of 3 decline over the case with cold initial conditions. No. 1, 2003 RESONANCE OCCUPATION IN THE KUIPER BELT 437

5 : 2 MMR would be reduced by 50% compared to that shown in Figure 9. The resultant population ratio between the 5 : 2 and 2 : 1MMRsof10 : 29 would differ only by a factor of 3 from the observed ratio of 7 : 6 if we include both DES and non-DES data sets. In summary, slow resonance sweeping over a primordial Kuiper belt that comprises both preheated orbits having i; ee0:2 and cold orbits having ed0:1 can populate the 2 : 1 and 5 : 2 MMRs with efficiencies that do not seem irreconcilable with the observations. The models can be tuned to match the observations by adjusting the initial eccentricity, inclination, and semimajor axis distributions of belt particles prior to the migration phase. We have not undertaken such tuning here; our main conclusion is that capture into the 5 : 2 MMR is made substantially more efficient and generates 5 : 2 resonant orbits similar to those observed by preheating the belt prior to resonance sweeping. Of course, our finding does not address the question of what was responsi- ble for this preheating.

3.2. Direct Scattering into the 5 : 2MMR Is it possible that KBOs in the 5 : 2 MMR may not have been captured via resonance sweeping, but were rather grav- itationally scattered into that resonance by close encounters Fig. 8.—Estimated width of the 5 : 2 MMR, derived by numerical integration of the circular, planar, restricted three-body model for the Sun/ with one or more massive objects? In a-e-i space, the prox- Neptune/KBO system. At each point on the above grid in a-e space, a test imity of our 5 : 2 resonant KBOs to orbits traditionally particle’s trajectory is integrated in the gravitational fields of the Sun and described as ‘‘ scattered ’’ suggests direct scattering by Neptune for 3 105 yr using a time step of 0.6 yr. Test particles share the Neptune as a population mechanism. same initial i ¼ 0, ¼ 0, ! ¼ =2, and M ¼ 0. Initial elements of Neptune To test this hypothesis, we integrate the trajectories of are given by a ¼ 30:1 AU, e ¼ i ¼ ¼ ! ¼ M ¼ 0. Thus, each N N N N N N 400 test particles on initially low-eccentricity, low-inclina- particle’s initial 5:2 ¼ . Plus signs denote particles for which 5:2 circu- lates, crosses denote particles that encounter the Hill sphere of Neptune, tion orbits in the vicinity of Neptune. The test particles’ and open squares denote particles that librate in the 5 : 2 MMR. The reso- semimajor axes, eccentricities, and inclinations range nant width is greatest, Da 0:8 AU, at ee0:2. Our three-body model is between 31.7 and 35.7 AU (2–7 Neptunian Hill radii from used only to generate this figure and for no other figure in this paper. Neptune’s semimajor axis), 0.00 and 0.02, and 0.00 and 0.01 rad, respectively. The other orbital angles are uniformly and randomly distributed between 0 and 2. The duration of the We have verified that these 29 objects are adiabatically cap- integration is 5 107 yr. No migration is imposed on any of tured by the sweeping 2 : 1 resonance and are not scattered the planets, whose initial positions and velocities are taken into it by close encounters with one of the giant planets. from Cohen, Hubbard, & Oesterwinter (1973). We again These captured particles originated on low-eccentricity employ the symplectic integrator, SyMBA; this integrator orbits, ed0:1. handles close encounters with better accuracy than does Taken at face value, one problem with the simulation swift_rmvs3. We adopt a time step in the absence of close depicted in Figure 9 is that it predicts a large proportion of encounters of 0.6 yr. nonresonant particles having semimajor axes between 50 Figure 10 summarizes the results of this simulation of and 55 AU that, to date, are not observed. The problem of direct scattering into MMRs. Of 400 test particles, zero, the ‘‘ Kuiper Cliff ’’—a sudden decrease in the surface den- one, two, and one particles are scattered into the 3 : 1, 5 : 2, sity of planetesimals outside 50 AU—has been discussed 2 : 1, and 3 : 2 MMRs, respectively. These resonant KBOs extensively in the literature (see, e.g., Jewitt, Luu, & Trujillo have substantial eccentricities, between 0.19 and 0.45. Thus, 1998; Gladman et al. 1998; Chiang & Brown 1999; Allen, the existence of resonant KBOs having high eccentricities, Bernstein, & Malhotra 2001; Trujillo & Brown 2001). We taken at face value, does not necessarily imply capture and regard the statistical significance of the observed edge of the adiabatic excitation by migratory resonances. belt as still marginal at best (see Allen et al. 2001). But even The relative capture efficiencies between the 3 : 1, 5 : 2, apart from possible observational selection biases, there are 2 : 1, and 3 : 2 MMRs in our direct scattering simulation do a number of factors that would help to improve the agree- not appear irreconcilable with the observations, given the ment between Figure 9 and observation. First, a fraction of small number statistics (both observationally and the simulated nonresonant objects between a ¼ 50 and 55 theoretically), observational biases (CJ), and uncertainties AU have large eccentricities and are not phase protected regarding actual initial conditions. from Neptune, so that they are unlikely to survive in their Direct scattering by Neptune, however, predicts libration current orbits for the age of the solar system. Second, if the amplitudes that are generally larger than those observed. Kuiper Cliff is real, we may impose an edge to our distribu- For our (four) simulated resonant particles inhabiting the tion at a ¼ 50 AU prior to resonance sweeping that would 5 : 2, 2 : 1, and 3 : 2 MMRs, libration amplitudes all exceed obviously reduce the number of objects in this region after 160. This is to be compared, for example, with the libration resonance sweeping. The number of objects caught in the amplitudes exhibited by our three observed 5 : 2 resonant 438 CHIANG ET AL. Vol. 126

Fig. 9.—Results of a migration simulation designed to gauge the capture efficiency of the 5 : 2 resonance under hot initial conditions. Left-hand panels display hot initial conditions of 400 test particles prior to sweeping by the 5 : 2 resonance, whose nominal location is indicated by the dashed line. Right-hand panels display the aftermath of resonance sweeping. Open diamonds denote librating particles in the 2 : 1, 5 : 2, and 3 : 1 MMRs. A total of 20 particles are adiabatically swept into the 5 : 2 MMR and have their eccentricities and inclinations excited above their initial values. A total of 29 particles are swept into the 2 : 1 resonance at a 47:8 AU. The libration amplitudes of the simulated 5 : 2 resonant KBOs range from 16 to 145, with six particles having amplitudes in the range between 90 and 140 (data not shown). The relative efficiencies of capture into the 2 : 1 and 5 : 2 MMRs, the predicted libration characteristics of 5 : 2 resonant particles, and the large final eccentricities and inclinations of resonant particles can all be reconciled with the observations, in contrast to the case using only cold initial conditions.

KBOs, which range between 90 and 140 (see Fig. 4). While into the 2 : 1 MMR librate with large amplitude about we cannot rule out the possibility that 1998WA31 represents h2:1i¼180 . This conflicts with the observed small libra- such a directly scattered, dynamically young object based tion amplitudes about h2:1i75 : the single confirmed on its unstable behavior on timescales of megayears to giga- Twotino in our survey librates with small libration ampli- years (see x 2.2), the small libration amplitudes of (38084) tude (50 ) about h2:1i88 , while four secure non-DES 1999HB12 and 2001KC77 do not match those predicted by Twotinos are characterized by h2:1i70 ,67 ,74, and the scattering simulation. (Further, as noted in x 2.2, it 83 (see CJ). Only one secure non-DES Twotino (2000JG81) remains possible that future improvements in the accuracy resembles a simulated particle, librating about h2:1i¼180 of the trajectory of 1998WA31 may cause it to join its more with an amplitude of 160 . Note that we interpret the stable brethren.) observed asymmetrically librating Twotinos to be primor- The problem of excessive libration amplitudes was dial residents, since they resemble the stable particles reported in a similar context by Levison & Stern (1995), simulated by Nesvorny & Roig (2001; see their x 3.4). who investigated ways to excite Pluto’s orbit to its present Is it possible that our simulation contains too few par- high eccentricity and inclination using only gravitational ticles to fully explore phase space and that we have been interactions with the giant planets in their current orbits. unlucky in the outcome of libration profiles? We do not Possible resolutions to this difficulty include invoking physi- believe so. Objects barely bound to MMRs are to be cal collisions with and/or gravitational scatterings off pri- expected from the direct scattering hypothesis because for mordial KBOs (Levison & Stern 1995). The disagreement Neptune to heat the orbit of a test particle significantly, the between predicted and observed libration characteristics distance of closest approach must be small, within several seems particularly severe for the 2 : 1 resonant objects. In Hill radii of Neptune. During the (brief) close encounter, our direct scattering simulation, the two particles scattered a particle’s velocity is radically altered, but because the No. 1, 2003 RESONANCE OCCUPATION IN THE KUIPER BELT 439

Fig. 10.—Results of a direct scattering simulation in which no migration is imposed on any of the planets. Left-hand panels display cold initial conditions for 400 test particles situated between 2 and 7 Neptunian Hill radii from Neptune. Right-hand panels display conditions after 50 Myr. One, two, and one particles, represented by open diamonds, are scattered directly into the 5 : 2, 2 : 1, and 3 : 2 MMRs, respectively. The eccentricities, inclinations, and relative numbers of these particles appear consistent with the observations. However, the predicted and observed libration characteristics disagree. The four simulated resonant particles have libration amplitudes between 160 and 175 (data not shown in this ﬁgure), too large compared to the moderate amplitudes observed.

particle’s position during the encounter is relatively for their hypothesized Neptune Trojans that about 50% of unchanged, subsequent close approaches between particle them survive for 4 Gyr in a postmigration solar system. and planet will occur at similarly small distances. Large Distinct from the retainment efficiency of the 1 : 1 MMR libration amplitude is synonymous with small distance of is the efficiency with which a migrating Neptune captures closest approach; thus, close encounters with Neptune alone objects into the 1 : 1 MMR; this capture efficiency has not are expected to yield only tenuously bound resonant par- been reported in the literature. To remedy this deficiency, ticles whose resonant locks are easily broken. we execute a migration simulation similar to the one described in x 3.1.1, except that the initial semimajor axes of 3.3. Neptune’s Migration and the 1 : 1MMR the 400 test particles are distributed between 24.1 AU (=1 Gomes (1998) explores, via numerical simulations similar AU greater than the initial location of the 1 : 1 MMR) and to those presented here, the ability of a migratory Neptune 29.1 AU (=1 AU less than the final location of the 1 : 1 to retain (as opposed to capture) a Trojan population. A MMR). Only cold initial conditions (initial 0 e 0:05, variety of migration histories for the giant planets are tested; 0 i 0:025) are employed; hot initial conditions would between 20% and 82% of his hypothetical Neptunian conflict with the observed low e and low i of 2001QR322. Trojans remain bound to the 1 : 1 MMR throughout the This is because, as described in greater detail below, migra- migration phase. He concludes that unless the migration tion is not expected to significantly alter the eccentricities history of the giant planets was such as to engender diver- and inclinations of Trojans. Results of the migration simu- gent resonance crossings and excitation of planetary eccen- lation are displayed in Figure 11. Of 400 particles poten- tricities to values of 0.1—a history that would, prima tially swept into the 1 : 1 MMR, no particle is captured. The facie, conflict with the small orbital eccentricities currently overwhelming fate of the particles is to be scattered to larger exhibited by Uranus and Neptune—a Neptunian Trojan semimajor axes, eccentricities, and inclinations. The low population might be expected to exist today. This finding is capture efficiency of d0.0025 suggests that Neptunian strongly reinforced by Nesvorny & Dones (2002), who find Trojans do not owe their existence to Neptune’s migration. 440 CHIANG ET AL. Vol. 126

Fig. 11.—Results of a migration simulation designed to gauge the capture eﬃciency of the 1 : 1 resonance. Left-hand panels display cold initial conditions of 400 test particles prior to sweeping by the 1 : 1 resonance, whose nominal location is indicated by the dashed line. Right-hand panels display the aftermath of resonance sweeping. No particle is captured into a Trojan-type orbit. Particles instead suﬀer close encounters with Neptune and are scattered onto highly eccentric and inclined orbits.

Our conclusion is subject to the caveat that we have not inappropriate for 2001QR322. Unlike the case of exterior modeled a possible stochastic component to Neptune’s resonances, outward migration causes the eccentricities and migration (see x 3.1.1 and the Appendix). inclinations of Trojans to decline. This behavior can be seen Capture scenarios that invoke frictional drag from solar from the adiabatic invariant, Cpq, associated with an MMR nebular gas, as well as damping of libration amplitudes via for which the ratio of mean orbital periods is p : ðp þ qÞ ( p gaseous envelope accretion by the host planet, are charac- and q are integers and q < 0 for exterior resonances): terized by healthy capture efficiencies for Jupiter’s Trojans pffiffiffiffiffiffiffiffiffiffihipffiffiffiffiffiffiffiffiffiffiffiffiffi 2 (Marzari & Scholl 1998; Peale 1993) but are expected to be Cpq ¼ Ma ðp þ qÞp 1 e cos i ð2Þ less efficient for Neptune’s Trojans. The decrease in effi- pffiffiffiffiffiffiffiffiffiffi M a ciency arises because Neptune’s hydrogen/helium compo- e2 þ i2 ð3Þ nent amounts to only 5%–20% of Neptune’s total mass, 2 while Jupiter’s hydrogen/helium component comprises (see, e.g., Yu & Tremaine 2001). For the last equality, we 90% of that planet’s mass (Lissauer 1993). We also note have taken p ¼ 1, q ¼ 0, and e; i5 1. Then as a increases, that frictional drag was likely to have been ineffective if e2 þ i2 must decrease. Fleming & Hamilton (2000) find in 2001QR322 possessed its current diameter (estimated 130– numerical simulations that, indeed, both eccentricity and 230 km, assuming a 12%–4% albedo) at the time of capture; inclination decrease as the semimajor axis increases, in 2001QR322 must have grown from the collisional agglomer- 8 quantitative agreement with the adiabatic invariant. The ation of smaller bodies that did feel gas drag (Peale 1993). effect is extremely weak. If we take i2 to be always compara- We now justify our earlier assertion that hot initial condi- 2 ble to e (as we can for the current orbit of 2001QR322), then tions prior to resonance capture into the 1 : 1 resonance are 1=4 ðef =eiÞðif =iiÞðai=af Þ . Migration scenarios adopt, for Neptune, 0:7dai=af 1; then the eccentricities and inclinations of its Trojans decline by at most 10% as a result 8 We note in passing that Neptune’s irregular satellite, Triton, might be thought of as inhabiting a 1 : 1 resonance with Neptune and might have of outward migration. Thus, the original eccentricity and been captured from an initially heliocentric orbit by colliding with an inclination of 2001QR322 cannot have exceeded their cur- ancient regular satellite of Neptune (Goldreich et al. 1989). rent (low) values by more than 10%. No. 1, 2003 RESONANCE OCCUPATION IN THE KUIPER BELT 441

Perhaps the most natural time for Neptune to accrue its about Neptune’s L4 and L5 points. For comparison, 10 Trojans is during its assembly from planetesimals. For mass Jovian Trojans exist having diameters between 100 and 200 accretion timescales that are long compared to the Trojan km (Davis et al. 2003). libration period, a 30-fold increase in the mass of the host Our DES has uncovered three members of the 5 : 2 planet converts co-orbital horseshoe-type orbits into tad- MMR. Among all resonant KBOs, these objects possess the pole-type orbits and reduces the libration amplitudes of highest orbital eccentricities and substantial orbital inclina- tadpole-type Trojans by factors of 2–3 (Marzari & Scholl tions. Their orbits cannot be a consequence of the standard 1998; Fleming & Hamilton 2000). model of Neptune’s migration; they cannot have originated from low-e, low-i orbits that underwent resonant capture and adiabatic excitation by a migratory Neptune. The prob- 4. SUMMARY AND DISCUSSION ability of resonant capture into the 5 : 2 MMR under cold We have presented, as part of our ongoing Deep Ecliptic initial conditions is too small compared with the probabil- Survey, the most complete picture of MMR occupation in ities of capture into the 2 : 1and3: 2 MMRs; the standard the Kuiper belt to date (Fig. 3). We have discovered mem- model predicts a population ratio of 0.01 between the 5 : 2 bers of the 5 : 2, 2 : 1, 7 : 4, 3 : 2, 4 : 3, and 1 : 1 resonances. and 2 : 1 MMRs that is not easily reconciled with the These KBOs represent secure identifications in the sense observed ratio of 3. Moreover, the orbital eccentricities that (1) their 1 fractional uncertainties in semimajor axis and inclinations of 5 : 2 resonant objects that are predicted lie between 3% and 0.003% and (2) numerical integrations by the standard migration model are too low compared with of orbit solutions distributed over the 1 confidence surface their observed values. of possible fitted orbits consistently yield resonant argu- The inability of the 5 : 2 MMR to capture objects on low- ments that librate for at least 3 Myr. In the special cases of eccentricity orbits is reflected in the vanishingly small width the 1 : 1 and 5 : 2 resonances, we have checked by explicit of the resonance at e ¼ 0. By contrast, we have found by numerical orbit integrations that orbit solutions distributed numerical simulations akin to those that produced Figure 8 over 5 confidence surfaces also yield libration for at least 3 that the 2 : 1 and 3 : 2 MMRs both widen as e decreases from Myr and that orbit solutions distributed over 1 confidence 0.05 to 0. Note that estimates by Malhotra (1995) of reso- surfaces yield libration for up to 1 Gyr for our 1 : 1 resonant nant widths do not extend to e < 0:05. Murray & Dermott object and for a subset of our 5 : 2 resonant objects. (1999; see their Fig. 8.7) discuss this qualitative difference Object 2001QR322, the first discovered Neptunian Trojan, between the low-eccentricity behavior of first-order interior librates about the leading Lagrange (L4) point of Neptune. MMRs and that of higher order interior MMRs. We reserve Numerical integrations of its trajectory that account for the a more detailed theoretical exploration of the dynamics and presence of the four giant planets reveal that libration per- capture efficiencies of exterior MMRs to a future study. sists for at least 1 Gyr. Furthermore, the orbital elements of The simplest channel for populating the 5 : 2 MMR with 2001QR322 and its small libration amplitude of 24 are con- objects like those that we have observed involves adiabati- sistent with the properties of Neptunian Trojans that are cally slow sweeping of that MMR over a preheated Kuiper stable for 4 Gyr, as described by, e.g., Nesvorny & Dones belt, i.e., one containing a significant proportion of initially (2002). It seems unlikely that the Trojan was captured into high-eccentricity (ee0:2), high-inclination (ie0:2) orbits the 1 : 1 MMR purely by dint of Neptune’s hypothesized prior to the migration phase. Capture efficiencies increase at migration; as Neptune encroaches on an object, the latter is large e for the 5 : 2 resonance, a reflection of the greater sooner scattered onto a highly eccentric and inclined orbit width of this resonance at large e and its vanishingly small than caught into co-orbital resonance. More probably, width at small e. Neptunian Trojans predate the migration phase and owe The libration amplitudes predicted by resonance sweep- their existence to the same process that presumably gave rise ing over a preheated belt are moderate, between 16 and to the Jovian Trojans: trapping of planetesimals into libra- 145, and accord well with those observed. tion about the L4/L5 points of an accreting protoplanetary Direct scattering of objects into the 5 : 2 and other reso- core (Marzari & Scholl 1998; Fleming & Hamilton 2000). nances via close encounters with Neptune can also generate Subsequent outward radial migration by Neptune due to the large eccentricities and inclinations that are observed the scattering of planetesimals causes 20%–82% of Neptu- but generally fails to reproduce the observed libration prop- nian Trojans to escape the resonance (Gomes 1998) and erties. Direct scattering yields objects that are barely bound decreases the eccentricities and inclinations of the remaining to MMRs; large libration amplitudes exceeding 160 are bound fraction by modest amounts, 10% at most. Taken predicted for the 5 : 2, 2 : 1, and 3 : 2 MMRs, in conflict with together, the long-term stability of 2001QR322 (this paper; the observations. Additional mechanisms—gravitational Nesvorny & Dones 2002) and its relative insensitivity to dra- interactions and/or physical collisions with bodies other matic changes in Neptune’s orbit lead us to regard Neptu- than Neptune—would need to be invoked to dampen libra- nian Trojans as dynamically pristine compared to the rest of tion amplitudes. Levison & Stern (1995) elaborate on a ser- the Kuiper belt. The 1 : 1 MMR acts as a shelter against ies of events that can dampen the libration amplitude of close encounters and dynamical excitation by resonance Pluto in the 3 : 2 resonance; analogous events would need to sweeping. It is possible that the Trojan has always remained be invoked to dampen the libration amplitudes of members confined to heliocentric distances of 20–30 AU. of other MMRs. These mechanisms require the ancient For an assumed albedo in the range of 12%–4%, our Kuiper belt to be orders of magnitude more populous than Neptune Trojan is 130–230 km in diameter. When our DES it is today, a prospect that by itself does not appear unrea- search fields are overlaid on model-dependent predictions sonable, given the requirements of planet formation models of the sky density of Neptunian Trojans constructed by (Kenyon 2002) and ongoing dynamical (Holman & Wisdom Nesvorny & Dones (2002), we estimate that between 20 1993; Duncan, Levison, & Budd 1995) and collisional and 60 Neptune Trojans resembling 2001QR322 librate erosion of the belt. 442 CHIANG ET AL. Vol. 126

Nonetheless, we feel that Occam’s razor, as well as the improved this paper. The NOAO observing facilities used physical plausibility and seeming inevitability of planetary in the Deep Ecliptic Survey are supported by the National migration driven by planetesimal scattering (Fernandez & Science Foundation. Ip 1984; Hahn & Malhotra 1999), would seem to disfavor resonance population mechanisms that do not invoke APPENDIX migration. Perhaps the principal objection to this mechanism lies in the possibility that Neptune’s migration was BREAKDOWN OF SMOOTH MIGRATION insufficiently smooth to resonantly capture objects; we dis- Here we crudely estimate the critical sizes of Neptune- cuss quantitatively this possibility in the Appendix. Chiang & Jordan (2002) offer a more objective test of the migra- encountering planetesimals above which our assumption of smooth migration would be invalid. We imagine that at tion hypothesis; for sufficiently fast migration rates, the each instant during Neptune’s migration, Neptune’s number of 2 : 1 resonant KBOs having libration centers sphere of influence—r m =3m 1=3a in extent, where h i270 should exceed those having h i90 by H ð N Þ N 2:1 2:1 m and m are the mass of Neptune and of the Sun, respec- factors of 3. An asymmetry in libration center populations N tively, and a is the semimajor axis of Neptune—contains N translates directly into an asymmetry in the sky density of N m Twotinos about the Sun-Neptune line. By contrast, if the planetesimals each having D mass. A typicalpffiffiffiffiffi N 2 : 1 resonance were populated by direct scattering, the Poisson fluctuation in the number of planetesimals is . It is this random fluctuation that generates a change of ran- libration centers would presumably be equally populated. dom sign in Neptune’s semimajor axis, by an amount of At present, the number (6) of known Twotinos is too small pffiffiffiffiffi a N m=m a to permit the drawing of firm conclusions. order ðD NÞ N after all the planetesimals within the sphere of influence are scattered away. The dura- In summary, it is most straightforward to reproduce the tion of encounter between each planetesimal and observed pattern of resonance occupation in the Kuiper belt P by presupposing both initially cold orbits (to populate reso- Neptune is of order Neptune’s orbital period, N. Then the magnitude of the random component of Neptune’s nances such as the 3 : 2and2: 1 MMRs) and initially hot migration rate is of order a/P . orbits (to populate resonances such as the 5 : 2 MMR) prior N to Neptune’s migration. For the migration to be smooth, What might have heated the primordial Kuiper belt prior a < a_ mean ; ðA1Þ to resonance sweeping? Models of planetesimal formation PN predict eccentricities and inclinations of less than 0.05 (see, e.g., Kenyon & Luu 1999; Lissauer 1993; Kokubo & where a_ mean is the mean (smooth) migration rate that arises Ida 1992), values below what are required to capture KBOs from the mean difference in Neptune-encountering planetsi- into the 5 : 2 resonance with the relative efficiencies mal fluxes having high specific angular momentum and observed. Thommes et al. (1999, 2002) propose that the 10 fluxes having low specific angular momentum compared to Neptune. In our simulations, a_ mean ¼ DaN expðt=Þ=, M embryonic cores of Neptune and Uranus were scattered 7 into the ancient belt and heated KBOs by dynamical fric- where DaN ¼ 7 AU and ¼ 10 yr. tion. A possible problem with this scenario is that Neptune’s We estimate the surface mass density of planetesimals orbital history may be too violent to generate a significant within the sphere of influence to be the surface mass density Trojan population. of planetesimals throughout the disk. Hahn & Malhotra By contrast, we believe there exists another preheating (1999) find in their numerical simulations of planetary mechanism that is more natural: scattering of KBOs to large migration (using effective particle sizes too large to engender e and large i by Neptune as that planet migrated outward smooth migration) that mdisk 50 M of material must be (Gomes 2002). The formation of a primitive scattered disk interspersed between a ¼ 10 AU and adisk ¼ 50 AU to drive that is later swept over by MMRs seems a natural conse- Neptune’s orbit outward by DaN 7 AU. Then our order- quence of planetary migration driven by planetesimal scat- of-magnitude estimate for the surface mass density every- where is NDm=r2 m =a2 . We solve this equation tering. What requires further elucidation is how the s disk disk for N and insert into equation (A1) to find perihelia of the scattered (preheated) objects are raised so as to avoid further scatterings by Neptune. Dm a 2 Da 2 P 2 m 2t < disk N N N exp : ðA2Þ We are indebted to Jana Pittichova for donating her tele- mN aN rH mdisk scope time to secure astrometric observations of our For mN ¼ 17 M, aN ¼ 25 AU, rH ¼ 0:6 AU, PN ¼ 130 yr, Neptune Trojan that helped to solidify its dynamical iden- t ¼ ¼ 107 yr, and other parameters as listed above, we find 9 tity. E. I. C. acknowledges support from National Science that Dm=mN < 4 10 for the migration to be smooth. Foundation Planetary Astronomy grant AST 02-05892, Spheres of density 2 g cm3 having radii less than 40 km Hubble Space Telescope Theory grant HST-AR-09514.01- would suffice. If Neptune’s migration instead occurred over A, and a Faculty Research grant awarded by the University timescales of ¼ 3 106 yr, the critical radius grows to 80 of California at Berkeley. M. W. B., R. L. M., and L. H. W. km. Kenyon (2002) calculates that 90% of the mass of the are supported in part by NASA grants NAG 5-8990 and primordial Kuiper belt was contained in bodies having radii NAG 5-11058. Research by J. L. E. and S. D. K. is sup- of 0.1–10 km at heliocentric distances of 40–50 AU. It is not ported, in part, by NASA grant NAG 5-10444. D. E. T. is clear, however, how these accretion calculations should be supported by grants from the American Astronomical modified at distances of 20–30 AU where Neptune resided. Society and the Space Telescope Science Institute. K. J. M. An inopportune encounter between Neptune and a single is supported by NASA grant NAG 5-4495. We thank Kelly massive planetesimal might have caused the former to lose Clancy and Mark Krumholz for assistance and David whatever retinue of resonant objects it had accumulated Nesvorny for a thoughtful referee’s report that significantly previously by smooth migration. No. 1, 2003 RESONANCE OCCUPATION IN THE KUIPER BELT 443

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